This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A ...
More

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.Less

Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Johns

Published in print: 2005-07-07

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.

This book interprets the book of nature for curious readers of all sorts, but especially for those looking to appreciate the power and beauty of physics without having to grapple with the ...
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This book interprets the book of nature for curious readers of all sorts, but especially for those looking to appreciate the power and beauty of physics without having to grapple with the mathematics. Hundreds of little diagrams take the place of the equations that physicists otherwise need to tell the tale. It is a tale, no less, of how the world is put together and how it works, of how the universe might have arisen and where it might be going.Less

Knowing : The Nature of Physical Law

Michael Munowitz

Published in print: 2006-01-12

This book interprets the book of nature for curious readers of all sorts, but especially for those looking to appreciate the power and beauty of physics without having to grapple with the mathematics. Hundreds of little diagrams take the place of the equations that physicists otherwise need to tell the tale. It is a tale, no less, of how the world is put together and how it works, of how the universe might have arisen and where it might be going.

This chapter discusses limiting correspondence relations between quantum mechanics and classical mechanics. The semiclassical limit is singular and no reductive relation obtains between these two ...
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This chapter discusses limiting correspondence relations between quantum mechanics and classical mechanics. The semiclassical limit is singular and no reductive relation obtains between these two theories. It is argued that the fundamental theory (quantum mechanics) is explanatorily deficient in that it cannot fully account for aspects of the asymptotic borderland between classical and quantum mechanics.Less

Intertheoretic Relations—Mechanics

Robert W. Batterman

Published in print: 2001-12-20

This chapter discusses limiting correspondence relations between quantum mechanics and classical mechanics. The semiclassical limit is singular and no reductive relation obtains between these two theories. It is argued that the fundamental theory (quantum mechanics) is explanatorily deficient in that it cannot fully account for aspects of the asymptotic borderland between classical and quantum mechanics.

The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. ...
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The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.Less

Hamilton–Jacobi Theory

Oliver Johns

Published in print: 2005-07-07

The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.

The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always ...
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The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always passing the same landmarks in the same sequence of times. There are no surprises in the macroscopic world of classical mechanics, a world where to know the present is to predict the future and retrace the past. Observations may differ superficially according to individual frames of reference, but there is always agreement on the larger issues. Energy is conserved. Momentum is conserved. Differences in coordinates and velocities, along with perceptions of motion and rest, are all reconciled, and Newton’s deterministic equations of motion are the law of the land.Less

Three-Part Invention

Michael Munowitz

Published in print: 2006-01-12

The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always passing the same landmarks in the same sequence of times. There are no surprises in the macroscopic world of classical mechanics, a world where to know the present is to predict the future and retrace the past. Observations may differ superficially according to individual frames of reference, but there is always agreement on the larger issues. Energy is conserved. Momentum is conserved. Differences in coordinates and velocities, along with perceptions of motion and rest, are all reconciled, and Newton’s deterministic equations of motion are the law of the land.

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical ...
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John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.Less

Alfonso Sorrentino

Published in print: 2015-05-26

John Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach—known as Aubry–Mather theory—singles out the existence of special orbits and invariant measures of the system, which possess a very rich dynamical and geometric structure. In particular, the associated invariant sets play a leading role in determining the global dynamics of the system. This book provides a comprehensive introduction to Mather's theory, and can serve as an interdisciplinary bridge for researchers and students from different fields seeking to acquaint themselves with the topic. Starting with the mathematical background from which Mather's theory was born, the book first focuses on the core questions the theory aims to answer—notably the destiny of broken invariant KAM tori and the onset of chaos—and describes how it can be viewed as a natural counterpart of KAM theory. The book achieves this by guiding readers through a detailed illustrative example, which also provides the basis for introducing the main ideas and concepts of the general theory. It then describes the whole theory and its subsequent developments and applications in their full generality.

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials ...
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This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials quadratic in the velocities. The important example of Hamiltonians quadratic in the momentum variables is first examined. In the simplest situations discussed in previous chapters, after an explicit integration over momenta p(t) one recovers the usual path integral. More general Hamiltonians are often met, for example, in the quantization of the motion on riemannian manifolds. The analysis is illustrated with the quantization of free motion on a sphere (or hypersphere) S N-1. A few relevant elements of classical mechanics are considered first.Less

PATH INTEGRALS IN PHASE SPACE

Jean Zinn-Justin

Published in print: 2004-11-18

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials quadratic in the velocities. The important example of Hamiltonians quadratic in the momentum variables is first examined. In the simplest situations discussed in previous chapters, after an explicit integration over momenta p(t) one recovers the usual path integral. More general Hamiltonians are often met, for example, in the quantization of the motion on riemannian manifolds. The analysis is illustrated with the quantization of free motion on a sphere (or hypersphere) SN-1. A few relevant elements of classical mechanics are considered first.

Polyatomic molecules created difficulties in kinetic theory from the start, with the problem of thermal capacities or specific heats. If the problem of specific heats was initially solved with the ...
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Polyatomic molecules created difficulties in kinetic theory from the start, with the problem of thermal capacities or specific heats. If the problem of specific heats was initially solved with the discovery of noble gases, the reconsideration of the old experiments on mercury vapour for monatomic gases, and Ludwig Boltzmann's suggestion to consider diatomic molecules as having five degrees of freedom, for more complex molecules the problem was bound to resurface later and to show up as one of the unsolved problems of classical mechanics. Before taking up this subject again, another difficulty which kinetic theory had to meet very soon is discussed. When Boltzmann wrote his equation, he included the case of polyatomic gases. He also made a remark on the importance of the extension of the H-theorem to this case. As a matter of fact, although the Boltzmann equation is correct, his original proof of the H-theorem for this case is not completely general, as pointed out for the first time by H.A. Lorentz in 1887.Less

The problem of polyatomic molecules

CARLO CERCIGNANI

Published in print: 2006-01-12

Polyatomic molecules created difficulties in kinetic theory from the start, with the problem of thermal capacities or specific heats. If the problem of specific heats was initially solved with the discovery of noble gases, the reconsideration of the old experiments on mercury vapour for monatomic gases, and Ludwig Boltzmann's suggestion to consider diatomic molecules as having five degrees of freedom, for more complex molecules the problem was bound to resurface later and to show up as one of the unsolved problems of classical mechanics. Before taking up this subject again, another difficulty which kinetic theory had to meet very soon is discussed. When Boltzmann wrote his equation, he included the case of polyatomic gases. He also made a remark on the importance of the extension of the H-theorem to this case. As a matter of fact, although the Boltzmann equation is correct, his original proof of the H-theorem for this case is not completely general, as pointed out for the first time by H.A. Lorentz in 1887.

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics, and of the relation of classical mechanics to relativity and quantum theory. A ...
More

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics, and of the relation of classical mechanics to relativity and quantum theory. A distinguishing feature is its integration of special relativity into the teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. The definition of canonical transformation no longer excludes the Lorentz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics since comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods such as linear vector operators and dyadics that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics require a rethinking of the quantum–classical connection.Less

Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Johns

Published in print: 2011-05-19

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics, and of the relation of classical mechanics to relativity and quantum theory. A distinguishing feature is its integration of special relativity into the teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. The definition of canonical transformation no longer excludes the Lorentz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics since comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods such as linear vector operators and dyadics that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics require a rethinking of the quantum–classical connection.

This chapter discusses a few of the central, but sometimes unspoken, scientific ideas or assumptions about the world. It first looks at the character of Newtonian physics and classical mechanics by ...
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This chapter discusses a few of the central, but sometimes unspoken, scientific ideas or assumptions about the world. It first looks at the character of Newtonian physics and classical mechanics by describing the work of Issac Newton. More specifically, it examines Newton's three laws of motion, expressed in his 1687 Principia Mathematica. The Principia not only launched the field of physics known as classical mechanics, but also paved the way for the emergence of science and the scientific method in the 1500s and 1600s. This scientific revolution was followed by the Enlightenment, an era characterised in part by advances in science. The determinism of the Newtonian universe was challenged by Pierre-Simon Laplace through his concept of free will.Less

Newton, Laplace, and Determinism

David P. Feldman

Published in print: 2012-08-09

This chapter discusses a few of the central, but sometimes unspoken, scientific ideas or assumptions about the world. It first looks at the character of Newtonian physics and classical mechanics by describing the work of Issac Newton. More specifically, it examines Newton's three laws of motion, expressed in his 1687 Principia Mathematica. The Principia not only launched the field of physics known as classical mechanics, but also paved the way for the emergence of science and the scientific method in the 1500s and 1600s. This scientific revolution was followed by the Enlightenment, an era characterised in part by advances in science. The determinism of the Newtonian universe was challenged by Pierre-Simon Laplace through his concept of free will.